What makes an Attack aircraft different from a Fighter?
Pew pew vs boom boom
Fighters primary mission is versus other aircraft. Attack aircraft’s primary mission is ground attack. There is some overlap depending on the aircraft.
. . . lately, not much in terms of the US AIr Force. IIRC, on their original introduction to the inventory, ‘Attack’ aircraft were primarily designated for Close Air Support. A-10s were for plinkin’ Soviet tanks, AV-8 Harriers were for USMC ground support, etc.
But over the course of the past twenty years, F-16s, F-18s, and F-15s have become so diversified in missions (evolving from air superiority fighters), that they took on attack roles as well.
Lines are sorta blurry anymore, especially with the F-117 having an 'F" designator, but acting pretty much as a stealth Attack bomber.
Tripler
I <3 A-10s & AH-64s.
It’s influenced by the fact that pilots consider it more prestigious to fly a fighter than a bomber or attack craft, and so most anything gets called a fighter.
The alphanumeric designator was an obvious clue as to the mission profile:
C=cargo and transport
B=bomber
P=patrol
H=helicopter
Fighters and Attack birds were both armed with guns and missiles…and as Tripler pointed out, mission profiles are blurred.
Shoot, I forgot to include the F-35, which aims to be a jack-of-all-trades, but master of absolutely none.
Tripler
But I’m totally not biased. . . 
This dichotomy was noticed when the DoD was trying to decide on how to brand the McDonnell-Douglas Hornet, the Navy’s newest (at the time) warplane.
It was going into fleet service as a replacement for the F-14 Tomcat, so obviously it should be the F-18. But Navy and Marine Corps attack squadrons would also be using it for maritime and ground attack duties, replacing the A-6 Intruder, so clearly it was going to be the A-18.
So they decided they’d call it the F/A-18 Hornet, because it’s two great tastes that taste great together.
The DoD didn’t decide to repeat that silly experiment with the fielding of the F-35 Lightning II. Even though it is used for exactly the same surface attack missions as the “A” series aircraft that went before it, it is still a Fighter, not an Attack plane.
In the modern Tri-Service designation system attack aircraft (“A” prefix) are primarily dedicated to performing a ground attack and/or tactical bombing role. Aircraft like the A-4 ‘Skyhawk’, A-7 ‘Intruder’, and A-10 ‘Thunderbolt’ were all designed and mostly used for this purpose.
Fighter aircraft (“F” prefix) can perform a variety of roles but are largely intended to perform some kind of air superiority role, either direct conflict (‘dogfighting’), tactical bombers, ‘missile trucks’ equipped with anti-radiation missiles to take out SAM sites and radars, or to perform bomber interception, among other roles.
Of course, there are often mixes, like the F/A-18 ‘Hornet’ and ‘Super Hornet’, which were designed from the ground up as multi-role aircraft, or the AC-130, which was originally a cargo aircraft but converted or built as a dedicated aerial gunship. And many aircraft are retasked to new missions as need arises or capability becomes apparent. The venerable F-15 was originally designed as an all-weather tactical fighter has served in ground attack, buddy refueling, and even as a carrier aircraft for the A-135 ASAT missile thanks to its high thrust-to-weight ratio and robust airframe.
As @Tripler notes, sometimes the designations are applied to obscure the intended mission role of the aircraft; the A-12 ‘Oxcart’ high speed reconnaissance plane was not and never intended to be a ground attack craft, and the F-117 ‘Nighthawk’ “stealth fighter” was actually a tactical bomber originally indented for carrying guided munitions and the variable yield variants of the B61 nuclear bomb.
I’m not sure where you got the notion that “most anything gets called a fighter” but it really isn’t true. Both the US Air Force and the Navy have far more support aircraft (the “C”, “E”, “H”, “K”, “R”, “U”, and others) than they do combat planes, and while it is true that flying these does not come with the celebrity of flying a fighter they are difficult jobs that are well regarded, especially tanker (“K”), reconnaissance (“R”), and search and rescue (“H”) where they may be in combat situations which are as much at risk as fighter aircraft.
That there are relatively few instances of dogfighting in the modern era of air combat involving jet fighters, especially in the post-Vietnam era is because there hasn’t been as much conflict, and nearly all air combat today between modern jets relies upon guided missiles because the speed of engagement and small munitions load makes direct air combat impractical. The dedicated air superiority fighter of WWII era tactics is an anachronism and frankly of limited utility in the European theatre, although naval aviation played a dramatic role in the Pacific theater of combat.
Stranger
A list of Military Aircraft and their respective designations:
List of United States Tri-Service aircraft designations - Wikipedia
Nitpick: The original F/A-18 was going into fleet service in the early 1980s to replace the A-7 Corsair II in light attack squadrons and F-4 Phantom IIs in fighter squadrons for carriers that couldn’t accommodate the F-14, namely the Midway and Coral Sea, not to replace either the F-14 or the A-6. It was only in the late 1990s when planned replacements for the F-14 Tomcat and A-6 Intruder fell through that the redesigned F/A-18E/F Super Hornet was chosen to replace the F-14 and A-6.
And for added confusion, the A-12 designation @Stranger_On_A_Train mentions which was a variant of the SR-71 was also shared by the intended replacement for the A-6, the McDonnell Douglas A-12 Avenger II.
I’ve wondered about that sometimes. Is a fighter harder to fly than a helicopter? As a non-pilot, ISTM a helicopter would take more skill than the other aircraft.
Traditional military aviation prestige has nothing to do with relative difficulty or operating skill.
Fighter pilots are “knights of the sky”. The others… just do utility stuff. To put it bluntly, at its extreme, commissioned peasants.
No, it’s not rational. Why would you expect that?
That is certainly the general public perception, and somewhat reflected within the military as well, possibly amplified by a certain movie about naval aviation involving a conflict of the US Navy against an unnamed adversarial in the Indian Ocean region (*cough* Sri Lanka *cough*). But attack, cargo, electronic surveillance, and other craft also have to land on aircraft carriers. The large cargo haulers (C-5, C-17, and C-141) had to land on runways carrying loads approaching their maximum design loads and in all manner of weather conditions. The lowly OV-10 ‘Bronco’ probably saw as much actual fire during Vietnam in its career of forward observer platform, reconnaissance, light attack, infil/exfil, and general flyabout as any ofter single designation except for the F-4 ‘Phantom’, and when calling for close air support what a solider, marine, or airman wants to hear is that a AC-130 ‘Spooky’ is en route.
Fighters are capable of being flown to the structural and mechanical limits, but are almost never subject to anything approaching those conditions outside of flight testing. The real challenge for fighter pilots isn’t actually the skill of flying the aircraft—which, since the advent of ‘fly by wire’ control systems is largely managed by the avionics that will largely prevent a pilot from inadvertently getting into trouble without a lot of effort—but in managing the workload of so many systems competing for their attention, especially in single pilot aircraft.
Stranger
Not mentioned yet: aircraft like the A-20 Havoc and the A-26 Invader that were definitely not “fighters”.
I don’t know about “knights”, but fighter jets are definitely the apex predators of the aircraft kingdom. They can shoot down anything in the air, and nothing else in the air can shoot them down. Right or wrong, that’s what makes them the most prestigious.
Except for the night-fighter versions of the A-20 (P-70 in USAAF service) which were. In fact they were called Havoc by the RAF to distinguish them from the B-20 Boston which was the bomber version.
About the only thing I think can be said with any actual authority is that any “B” or “A” designated aircraft isn’t likely to be tasked with fighting other airplanes.
But ones designated with an “F” may or may not be. It doesn’t necessarily indicate that they’re dedicated air-superiority fighters like the F-22, but nor is it typically used for planes only used for ground attack/interdiction. There are exceptions- the F-111 was a medium range attack plane that ended up with the “F” designation because of development shenanigans, and the F-117 was the “stealth fighter” that ended up with the “F” because of reasons that are still unclear, but likely revolve around maintaining secrecy and obscuring the nature of the plane.
Well, “shenanigans” started out as a naively well-intentioned desire for a multi-role aircraft. The Air Force wanted a medium bomber. The Navy wanted a fleet air defense fighter (essentially, a long-range air-to-air missile platform).
The Navy successfully fought off Defense Secretary Robert Macnamara’s mandate that both missions would be accomplished with one new aircraft but some of the technological advancements from the F-111B (the Navy variant) went on to figure prominently in the airplane the Navy eventually wanted and got: the F-14.
The most ridiculous was when the Raptor was briefly classified as “F/A-22” despite it being almost as far from an attack warplane as could be.
At one time P stood for ‘Pursuit’, later changed in the late '40s to F for Fighter. The Lockheed P-80 became the F-80 and the surviviing P-51s became F-51s.
The Navy had its own systems of aircraft designations, none of which resembled the Army Air Corp’s system.